Abstract: Fractal structures pervade nature and are receiving increasing engineering
attention towards the realization of broadband resonators and antennas. We show
that fractal resonators can support the emergence of high-dimensional chaotic
dynamics even in the context of an elementary, single-transistor oscillator
circuit. Sierpi\'nski gaskets of variable depth are constructed using discrete
capacitors and inductors, whose values are scaled according to a simple
sequence. It is found that in regular fractals of this kind each iteration
effectively adds a conjugate pole/zero pair, yielding gradually more complex
and broader frequency responses, which can also be implemented as much smaller
Foster equivalent networks. The resonators are instanced in the circuit as
one-port devices, replacing the inductors found in the initial version of the
oscillator. By means of a highly simplified numerical model, it is shown that
increasing the fractal depth elevates the dimension of the chaotic dynamics,
leading to high-order hyperchaos. This result is overall confirmed by SPICE
simulations and experiments, which however also reveal that the non-ideal
behavior of physical components hinders obtaining high-dimensional dynamics.
The issue could be practically mitigated by building the Foster equivalent
networks rather than the verbatim fractals. Furthermore, it is shown that
considerably more complex resonances, and consequently richer dynamics, can be
obtained by rendering the fractal resonators irregular through reshuffling the
inductors, or even by inserting a limited number of focal imperfections. The
present results draw attention to the potential usefulness of fractal
resonators for generating high-dimensional chaotic dynamics, and underline the
importance of irregularities and component non-idealities.